Discrete Risk Statistic Calculations

A number of statistics can be used to describe the sensitivity of the instrument's price to a change in the yield. The standard measures of Macaulay's duration, modified duration, convexity, and PVBP (present value of a basis point), are presented.

All of these measures typically rely on either the first or second derivatives of the instrument price with respect to yield (or equivalent), however due to the complexities of pricing many instruments, such derivatives need to be approximated.

To approximate a risk statistic, the instrument is required to be revalued with a minor increment in the yield. The resulting change in price over the change in yield is then used in the risk statistic calculations.

Duration is defined as the price elasticity of the instrument, that is, duration describes the percentage change in the price for a given percentage change in yield.

Note that because instrument price is a negative function of yield, the first derivative of price with respect to yield will also be negative, and the negative sign is therefore used in the above equation to ensure that the duration is always expressed as a positive number.

2. Modified Duration

Modified duration is the percentage change in the instrument price for a given change in yield, and is calculated as:

As with Macaulay duration, modified duration is always expressed as a positive number even though the relationship between yield and prices is negative.

3. Convexity

Convexity is used to approximate the percentage change in the instrument price that is not explained by the duration measure (resulting from the curvature in the instrument price and yield relationship). Convexity is defined as follows:

4. PVBP

The price value of a basis point change for every $100 face value can be determined as:

As with the duration statistics, PVBP is always expressed as a positive number.